STOCHASTIC

CONSUMPTION AND PORTFOLIO RULES 395
optimal consumption and portfolio rules corresponding to (48) and (49)
are
C*(rt = - - t L-h. (70)
C v > 8(1 - exp[(P - yv)(t - D/8]) p { ' V)
and
W.W= ^ A {w+ m ~r e "" T)) ) + { -^P « - «"-")• (71)
Comparing (70) and (71) with (48) and (49), one finds that, in computing
the optimal decision rules, the individual capitalizes the lifetime flow of
wage income at the market (risk-free) rate of interest and then treats the
capitalized value as an addition to the current stock of wealth. 23
The introduction of a stochastic wage income will cause increased
computational difficulties although the basic analysis is the same as for
the no-wage income case. For a solution to a particular example of a
stochastic wage problem, see example two of Section 8.
8. POISSON PROCESSES
The previous analyses always assumed that the underlying stochastic
processes were smooth functions of Brownian motions and, therefore,
continuous in both the time and state spaces. Although such processes
are reasonable models for price behavior of many types of liquid assets,
they are rather poor models for the description of other types. The
Poisson process is a continuous-time process which allows discrete (or
discontinuous) changes in the variables. The simplest independent Poisson
process defines the probability of an event occuring during a time interval
of length h (where h is as small as you like) as follows:
prob{the event does not occur in the time interval (/, t -f A)}
= 1 - A/i + 0(h),
prob{the event occurs once in the time interval (/, t + h)} (72)
= Xh + 0(/i),
prob{the event occurs more than once in the time interval
(r, t + /;)} = 0(h),
23 As Hakansson [6] has pointed out, (70) and (71) are consistent with the Friedman
Permanent Income and the Modigliani Life-Cycle hypotheses. However, in general,
this result will not hold.
THE CAPITAL GROWTH CRITERION AND CONTINUOUS-TIME MODELS